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We look for an enhancement of the correspondence model of peridynamics, emphasizing the elimination of zero-energy deformation modes. We propose an approach based on the notion of sub-horizons. The most useful feature of this proposal is the setup which, whilst providing solutions with the necessary stability, deviates only marginally from the original correspondence formulation. A thorough analysis of the sub-horizon based method is furnished based on the well-posedness of integral equations and energy spectrum, which clearly demonstrate a removal of zero energy modes. We also show how other forms of unphysical deformation modes, e.g. material collapse within horizon, jump discontinuities and vanishing energy modes, can be prevented with the present proposal. Finally, a set of numerical simulations are undertaken that attest to the remarkable efficacy of the sub-horizon based approach. (C) 2018 Elsevier B.V. All rights reserved.

We devise and evaluate numerically a Hybrid High-Order (HHO) method for incremental associative plasticity with small deformations. The HHO method uses as discrete unknowns piecewise polynomials of order $k\ge1$ on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The HHO method supports polyhedral meshes with non-matching interfaces, is free of volumetric-locking and the integration of the behavior law is performed only at cell-based quadrature nodes. Moreover, the principle of virtual work is satisfied locally with equilibrated tractions. Various two- and three-dimensional test cases from the literature are presented including comparison against known solutions and against results obtained with an industrial software using conforming and mixed finite elements., Comment: 28 pages, 26 figures, preprint

This paper develops a parametrically homogenized continuum damage mechanics (PHCDM) model for unidirectional fiber-reinforced composites undergoing progressive damage. The PHCDM models are designed to overcome limitations of prohibitive computational overhead associated with many homogenization methods. They are thermodynamically consistent, reduced-order continuum models with explicit representation of microstructural morphology. The PHCDM model is derived from detailed micromechanics of representative volume element (RVE) using energy equivalence principles. Micromechanical failure is due to fiber–matrix interface debonding and matrix cracking. The macroscopic PHCDM models represent damage anisotropy through a second-order damage tensor that contributes to the evolution of a damage surface in the space of damage work conjugate. The damage surface characterizes the initiation and evolution of damage. The constitutive relation between damage and its work conjugate is represented by an anisotropic fourth-order damage surface tensor \(P_{ijkl}\), whose components are expressed as functions of current damage state and composite morphology. These are calibrated and validated from homogenized micromechanical (HMM) responses. The PHCDM model is incorporated in a commercial finite element code, and analysis of macroscopic composite components is executed for understanding concurrent damage at multiple material length scales.

A superconvergent isogeometric formulation is presented to accurately analyze the natural frequencies for elastic continua. This formulation is realized by a set of superconvergent quadrature rules which are designed for the numerical integration of isogeometric mass and stiffness matrices. In order to obtain these quadrature rules, a natural frequency error measure for elastic continua is systematically deduced using the quadratic basis functions, where the mass and stiffness matrices are formulated by the assumed quadrature rules. In contrast to the quadrature-based superconvergent isogeometric formulation for the scalar-valued wave equations, it is shown that herein different quadrature rules are required for the mass and stiffness matrices to achieve the superconvergence of natural frequency computation for the vector-valued elastic continuum problems . Consequently, the superconvergent quadrature rules are established through optimizing the natural frequency accuracy. It turns out that with these quadrature rules, the accuracy of natural frequencies for elastic continua is improved by two orders compared with the standard isogeometric formulation employing the consistent mass matrix . Meanwhile, it is found that the superconvergent quadrature rules involve the wave propagation angle and consequently simplified quadrature rules without the angle dependence are further proposed for straightforward practical applications. Numerical results reveal the superconvergence of the proposed method regarding the natural frequencies for elastic continua.

Fracture of viscoelastic solids plays an important role in many applications but it is not yet well understood. In addition to the time and rate dependent response of viscoelastic materials, fracture of these solids is governed by nonlinear processes at the fracture process zone and could also be accelerated by viscous energy dissipation . To this end, we propose a new phase field formulation in which fracture of viscoelastic solids is driven by both elastic and viscous components of the energy. The formulation requires a single additional parameter to quantify the portion of the viscous energy that contributes to fracture and is shown to be thermodynamically consistent. Viscoelastic material behavior , in the form of a Generalized Maxwell model , is obtained through a standard Prony-series type expansion. Fracture driven by viscous dissipation is studied on several important benchmark problems, including (i) a bar under creep, relaxation, strain rate and cyclic loadings, and (ii) two 3-point asphalt-beam bending problems that lead to crack propagation under mode I and mixed mode conditions. It is shown that at low strain rates viscous dissipation accelerates the fracture growth rate but essentially does not affect the crack path, while at high rates the effect of viscous dissipation is minor.

This paper studies concurrent two-scale design optimization of composite structures filled with multiple microstructural unit cells. The task of the design problem is to simultaneously optimize microstructural configurations of the unit cells and their spatial distribution in the macroscale. To this end, a new topology optimization framework based on combined topology representation of the density model and the level set model is proposed. The homogenization method is used to link the material microstructural design and the macroscale design by evaluating the effective properties of the microstructures. In the microscale, topology optimization of multiple microstructural unit cells is performed with the density-based method. In the macroscale design, the distribution of multiple microstructural unit cells is optimized by the velocity field level set method, which inherits advantages of the implicit geometrical representation of the conventional level set model (relatively clear and smooth material boundaries/interfaces, more natural description of topological evolution). Moreover, the velocity field level set method maps the variational boundary shape optimization problem into a finite-dimensional design space, thus making it relatively easy and efficient to employ general mathematical programming algorithms to handle the multiple constraints and two types of design variables in the concurrent two-scale design problem. Numerical examples show that the present concurrent two-scale design method can generate meaningful designs of hierarchical cellular structures with well-defined boundaries and material interfaces.

The contribution is concerned with a numerical element formulation in boundary representation. It results in a polynomial element description with an arbitrary number of nodes on the boundary. Scaling the boundary description determines the interior domain. The scaling approach is adopted from the so-called scaled boundary finite element method (SBFEM), which is a semi-analytical formulation to analyze problems in linear elasticity . Within this method, the basic idea is to scale the boundary with respect to a scaling center. The boundary, which is denoted as circumferential direction , and the scaling direction span the parameter space. In the present approach, interpolations in scaling direction and circumferential direction are introduced. The interpolation in circumferential direction is independent of the scaling direction. The formulation is suitable to analyze problems in nonlinear solid mechanics. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located at the interior domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows per definition for Voronoi meshes and quadtree mesh generation . Numerical examples give rise to the performance of the present approach in comparison to other polygonal element formulations, like the virtual element method (VEM). Some benchmark tests show the capability of the element formulation. A comparison to standard and mixed element formulations is presented. The present approach is perfectly suitable to model heterogeneous structures with inclusions and voids. It avoids also staircase approximation of curved boundaries.

Isogeometric analysis (IGA) has been a particularly impactful development in the realm of Kirchhoff–Love thin-shell analysis because the high-order basis functions employed naturally satisfy the requirement of C 1 continuity. Still, engineering models of appreciable complexity, such as wind turbine blades, are typically modeled using multiple surface patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. A penalty approach for coupling adjacent patches is therefore presented. The proposed method imposes both displacement and rotational continuity and is applicable to either smooth or non-smooth interfaces and either matching or non-matching discretization. The penalty formulations require only a single, dimensionless penalty coefficient for both displacement and rotation coupling terms, alleviating the problem-dependent nature of the penalty parameters. Using this coupling methodology, numerous benchmark problems encapsulating a variety of analysis types, geometrical and material properties, and matching and non-matching interfaces are addressed. The coupling methodology produces consistently accurate results throughout all tests. Furthermore, the suggested penalty coefficient of α = 1 0 3 is shown to be effective for the wide range of problem configurations addressed. Finally, a realistic wind turbine blade model, consisting of 27 patches and 51 coupling interfaces and having a chordwise- and spanwise-variant composite material definition, is subjected to buckling, vibration, and nonlinear deformation analyses using the proposed approach.

A node-to-node (NTN) scheme for modeling three-dimensional contact problems within a scaled boundary finite element method (SBFEM) framework is proposed. Polyhedral elements with an arbitrary number of faces and nodes are constructed using the SBFEM. Only the boundary of the polyhedral element is discretized to accommodate a higher degree of flexibility in mesh transitioning. Nonmatching meshes can be simply converted into matching ones by appropriate node insertions, thereby allowing the use of a favorable NTN contact scheme. The general three-dimensional frictional contact is explicitly formulated as a mixed complementarity problem (MCP). The inherent nonlinearity in the three-dimensional friction condition is treated accurately without requiring piecewise linearization . Contact constraints for non-penetration and stick-slide are enforced directly in a complementarity format. Numerical examples with 1st and 2nd order elements demonstrate the accuracy and robustness of the proposed scheme.

In the present study, a geometrically nonlinear theory for circular cylindrical shells made of incompressible hyperelastic materials is developed. The 9-parameter theory is higher-order in both shear and thickness deformations. In particular, the four parameters describing the thickness deformation are obtained directly from the incompressibility condition. The hyperelastic law selected is a state-of-the-art material model in biomechanics of soft tissues and takes into account the dispersion of collagen fiber directions. Special cases, obtained from this hyperelastic law setting to zero one or some material coefficients, are the Neo-Hookean material and a soft biological material with two families of collagen fibers perfectly aligned. The proposed model is validated through comparison with the exact solution for axisymmetric cylindrical deformation of a thick cylinder. In particular, the shell theory developed herein is capable to describe, with extreme accuracy, even the post-stability problem of a pre-stretched and inflated Neo-Hookean cylinder until the thickness vanishes. Comparison to the solution of higher-order shear deformation theory, which neglects the thickness deformation and recovers the normal strain from the incompressibility condition, is also presented.

High-order spectral element methods (SEM) for large-eddy simulation (LES) are still very limited in industry. One of the main reasons behind this is the lack of robustness of SEM for under-resolved simulations, which can lead to the failure of the computation or to inaccurate results, aspects that are critical in an industrial setting. To help address this issue, we introduce a non-modal analysis technique that characterizes the numerical diffusion properties of spectral element methods for linear convection–diffusion problems, including the scales affected by numerical diffusion and the relationship between the amount of numerical diffusion and the level of under-resolution in the simulation. This framework differs from traditional eigenanalysis techniques in that all eigenmodes are taken into account with no need to differentiate them as physical or unphysical. While strictly speaking only valid for linear problems, the non-modal analysis is devised so that it can give critical insights for under-resolved nonlinear problems. For example, why do SEM sometimes suffer from numerical stability issues in LES? And, why do they at other times be robust and successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these questions in turn provides crucial guidelines to construct more robust and accurate schemes for LES. For illustration purposes, the non-modal analysis is applied to the hybridized discontinuous Galerkin methods as representatives of SEM. The effects of the polynomial order, the upwinding parameter and the Péclet number on the so-called short-term diffusion of the scheme are investigated. From a non-modal analysis point of view, and for the particular case of hybridized discontinuous Galerkin methods, polynomial orders between 2 and 4 with standard upwinding are found to be well suited for under-resolved turbulence simulations. For lower polynomial orders, diffusion is introduced in scales that are much larger than the grid resolution. For higher polynomial orders, as well as for strong under/over-upwinding, robustness issues can be expected due to low and non-monotonic numerical diffusion. The non-modal analysis results are tested against under-resolved turbulence simulations of the Burgers, Euler and Navier–Stokes equations. While devised in the linear setting, non-modal analysis successfully predicts the behavior of the scheme in the nonlinear problems considered. Although the focus of this paper is on LES, the non-modal analysis can be applied to other simulation fields characterized by under-resolved scales., United States. Air Force. Office of Scientific Research (Grant FA9550-16-1-0214)

A quadrilateral bi-cubic G 1 -conforming finite element for the analysis of Kirchhoff plates is presented. The rational version of the Gregory patch proposed by Loop et al. (2009) is the starting point of our formulation. This version of the Gregory patch consists in rational enhancement of the bi-cubic Bezier interpolation representing a suitable tool for designing G 1 -conforming quadrilateral element on C 0 -conforming un-structured meshes. The element includes as additional degrees of freedom the edge rotations like in the Loof-formulations but is only displacement based. Because of the presence of the rational functions, the second derivatives of the interpolation present a finite discontinuity at the corners of the element, that prevent the element from passing the bending patch test. The element so formulated does not present optimal rate of convergence under h -refinement operation. The formulation is enhanced enforcing these discontinuities to be zero by means of Lagrange multipliers. It is shown that with these constraints the element passes the patch test and presents optimal rate of convergence for unstructured mesh. In this way the rational conforming approximation collapses into a conforming re-arrangement of the complete bi-cubic Bezier interpolation. Some examples and benchmarks are presented in order to test the performance of the element for the Kirchhoff plate model.

For Kirchhoff–Love shell problems a new mixed formulation solely based on standard H 1 spaces is presented. This allows for flexibility in the construction of discretization spaces, e.g., standard C 0 -coupling of multi-patch isogeometric spaces is sufficient. In terms of solution strategies, efficient methods for standard second-order problems like multigrid methods can be used as building blocks of preconditioners for iterative solvers. Furthermore, a combination of the proposed mixed formulation of the bending part with a popular mixed formulation of the membrane part in order to avoid membrane locking is considered. The performance of both mixed formulations is demonstrated by numerical benchmark studies.

This work is concerned with the development of a novel model order reduction technique for FFT solvers. The underlying concept is a compressed sensing technique which allows the reconstruction of highly incomplete data using non-linear recovery algorithms based on convex optimization , provided the data is sparse or has a sparse representation in a transformed basis. In the context of FFT solvers, this concept is utilized to identify a reduced set of frequencies on a sampling pattern in the frequency domain based on which the Lippmann–Schwinger equation is discretized. Classical fixed-point iterations are performed to solve the local problem. Compared to the unreduced solution, a significant speed-up in CPU times at a negligibly small loss of accuracy in the overall constitutive response is observed. The generation of highly resolved local fields is easily possible in a post-processing step using reconstruction algorithms which are available as open source routines. The developed reduction technique does not require any time-consuming offline computations, e.g. for the generation of snapshots, is not restricted to any kinematic or constitutive assumptions and its implementation is straightforward. Composites consisting of elastic inclusions embedded in an i) elastic and ii) elastoplastic matrix are investigated as representative simulation examples.

Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space. However, in several cases it can further cause conditioning problems for which a number of remedies have been proposed in the framework of the extended/generalized finite element method (XFEM/GFEM). Those solutions often involve significant modifications to the initial method and result in increased implementation complexity. In the present work, a simple procedure for the local near-orthogonalization of enrichment functions is introduced, which significantly improves the conditioning of the resulting system matrices, while requiring only minor modifications to the initial method. Although application to different types of enrichment functions is possible, the resulting scheme is specialized for the singular enrichment functions used in linear elastic fracture mechanics and tested through benchmark problems.

This paper is concerned with the analysis of coupled mixed finite element methods applied to the Biot’s consolidation model. We consider two mixed formulations that use the stress tensor and Darcy velocity as primary variables as well as the displacement and pressure. The first formulation is with a symmetric stress tensor while the other enforces the symmetry of the stress weakly through the introduction of a Lagrange multiplier. The well-posedness of the two formulations is shown through Galerkin’s method and suitable a priori estimates. The two formulations are then discretized with the backward Euler scheme in time and with two mixed finite elements in space. We present next a general and unified a posteriori error analysis which is applicable for any flux- and stress-conforming discretization. Our estimates are based on H 1 ( Ω ) -conforming reconstruction of the pressure and a suitable H 1 ( Ω ) d -conforming reconstruction of the displacement; both are continuous and piecewise affine in time. These reconstructions are used to infer a guaranteed and fully computable upper bound on the energy-type error measuring the differences between the exact and the approximate pressure and displacement. The error components resulting from the spatial and the temporal discretization are distinguished. They are then used to design an adaptive space–time algorithm. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm.

Using a path-following algorithm to analyze a quasi-static nonlinear structural problem involves selecting an appropriate constraint function. This function should improve the desired performance targets of the path-following algorithm such as robustness, speed, accuracy, and smoothness. In order to be able to draw a fair objective selection of a constraint function, it is necessary to collect adequate constraint equations as well as to define the performance of nonlinear methods. In this paper, three new path-following constraints applicable for damage analysis of quasi-brittle materials are proposed. Additionally, performance criteria and their numerical measures for a posteriori assessment of robustness, smoothness, accuracy, and speed of solving nonlinear problems by a path-following method are proposed. Based on the proposed criteria, the performance of the three new constraints and two existing ones is compared for two example problems. As a result, the performance measures are shown to possess an ability to clearly explore the strengths of each constraint. They establish a firm basis for the assessment of not only path-following methods but also other methods for solving nonlinear structural problems.

Computing mode-decomposed energy release rates in arbitrarily shaped delaminations involving large fracture process zones has not been previously investigated. The J -integral is a suitable method for calculating this, because its domain-independence can be employed to reduce the integration domain to a cohesive interface, and reduce it to a line integral. However, the existing formulations for the evaluation of the mode-decomposed J -integrals rely on the assumption of negligible fracture process zones. In this work, a method for the computation of the mode-decomposed J -integrals in three-dimensional problems involving large fracture process zones and using the cohesive zone model approach is presented. The formulation is applicable to curved fronts with non-planar crack faces. A growth driving direction criterion, which takes into account the loading state at each point, is used to render the integration paths and to decompose the J -integral into loading modes. This results in curved and non-planar integration paths crossing the cohesive zone. Furthermore, its implementation into the finite element framework is also addressed. The formulation is validated against virtual crack closure technique (VCCT) and linear elastic fracture mechanics (LEFM)-based analytical solutions and the significance and generality of the formulation are demonstrated with crack propagation in a three-dimensional composite structure.

An automated computational framework is introduced to link the microstructure to the micromechanical behavior of a heterogeneous adhesive composed of an epoxy matrix and embedded silica particles. A new reconstruction algorithm is employed to synthesize 3D periodic microstructural models of this materials system based on morphological and statistical data extracted from micro-computed tomography images. A non-iterative mesh generation algorithm is implemented in parallel to transform resulting virtual microstructures into finite element (FE) models. The continuum ductile and cohesive damage models used for simulating the adhesive’s failure response are calibrated with experimental data and the statistical representative volume element (SRVE) is identified. We have then carried out several high-fidelity FE simulations to investigate the effects of pre-existing voids in the adhesive layer, surface roughness of adherends, and the bonding strength along particles–matrix interfaces on the failure response of the adhesive subject to tensile, compressive, and shear loads.

Nanowire based semiconductors are promising for nanogenerators. However, there exist limited numerical tools to analyze these type of structures taking into account effects which are of particular importance at nanoscale . Therefore, we present a finite deformation NURBS based formulation to model a multifunctional material that couples strain, strain gradient , polarization and free charge carriers simultaneously. Specifically, the weak form and consistent linearization of the piezoelectric semiconductor including flexoelectricity and non-local elasticity are introduced. The nonlinear equations are then discretized and solved by utilizing isogeometric analysis (IGA) which fulfills the C 1 continuity requirement. Several numerical examples are performed to investigate the influence of flexoelectricity and non-local elasticity in ZnO piezoelectric semiconductor nanowires under large deformation . The formulation developed in this work can contribute to the development of novel nanoelectromechanical coupling devices such as flexoelectric nanogenerators.

The contribution herein proposes an isogeometric generalized n th order perturbation-based stochastic method for exactly modeling/representing composite structures comprising of different materials with particular attention to both static and dynamic analysis of structures with random material characteristics. Herein, we exactly represent the geometric model via isogeometric analysis (IGA), such as exactly modeling the interfaces in composite structures with dissimilar materials and also continuous variable thicknesses that cannot be achieved by existing traditional methods such as FEM. Besides, only a limited or scant work has been conducted thus far with IGA and stochastic methods. We consider the uncertainties of elastic modulus and mass density into account as stochastic inputs in static and dynamic isogeometric stochastic analyses. Moreover, we derive and expand the IGA based random-input parameter and all state functions included in static and dynamic equilibrium equations around their expectations via a generalized n th order Taylor series using a small perturbation parameter e . In addition, we determine the probabilistic moments of the stochastic solution that satisfy the given accuracy requirement by expanding to n th order. The results obtained by the proposed method, the finite element method (wherever feasible), and Monte Carlo simulations for both benchmark and engineering applications verify the following: (a) the proposed methodology can achieve more accurate deterministic solutions with improved efficiency, thereby strengthening the effectiveness and efficiency brought by the stochastic method. This is in contrast to the FEM based method which weakens them, (b) on the other hand, it can more efficiently acquire reliable and more accurate stochastic results, both for expected values and standard deviations in the static and dynamic (free vibration) analyses. Note that the larger the problem size is, the more efficient the proposed method will be.

We propose a parallel and computationally efficient multi-split XFEM approach for 3-D analysis of heterogeneous materials. In this approach, multiple discontinuities (pores and reinforcement particles) may intersect any given element (we call those elements multi-split elements). These discontinuities are modeled by imposing additional degrees of freedom at the nodes. The main advantage of the proposed scheme is that the mesh size remains independent of the relative distance among the heterogeneities/discontinuities. The pores and reinforcement particles are assumed to be spherical. The simulations are performed for uniform and non-uniform heterogeneity distribution. The Young’s modulus of the heterogeneous material is evaluated for different amount of pores and reinforcement particles. To demonstrate the computational efficiency of the multi-split XFEM, elastic damage analysis is performed for the unit cell with 5% pores and 5% reinforcement particles under uniaxial tensile loading. These simulations show that the Young’s modulus decreases linearly with the increase in the volume fraction of the pores and increases linearly with the increase in volume fraction of reinforcement particles. The multi-split XFEM is found to be at least 1.8 times computationally efficient than standard XFEM and at least 6.7 times computationally efficient than FEM.

In this paper we present a new multi-resolution parallel framework, which is designed for large-scale SPH simulations of fluid dynamics. An adaptive rebalancing criterion and monitoring system is developed to integrate the CVP partitioning method as rebalancer to achieve dynamic load balancing of the system. A localized nested hierarchical data structure is developed in cooperation with a tailored parallel fast-neighbor-search algorithm to handle problems with arbitrarily adaptive smoothing-length and to construct ghost buffer particles in remote processors. The concept of “diffused graph” is proposed in this paper to improve the performance of the graph-based communication strategy. By utilizing the hybrid parallel model, the framework is able to exploit the full parallel potential of current state-of-the-art clusters based on Distributed Shared Memory (DSM) architectures. A range of gas dynamics benchmarks are investigated to demonstrate the capability of the framework and its unique characteristics. The performance is assessed in detail through intensive numerical experiments at various scales.

We discuss the implementation details and the numerical performance of the recently introduced nonconforming Trefftz virtual element method (Mascotto et al., 2018) for the 2D Helmholtz problem. In particular, we present a strategy to significantly reduce the ill-conditioning of the original method; such a recipe is based on an automatic filtering of the basis functions edge by edge, and therefore allows for a notable reduction of the number of degrees of freedom. A widespread set of numerical experiments, including an application to acoustic scattering, the h -, p -, and h p -versions of the method, is presented. Moreover, a comparison with other Trefftz-based methods for the Helmholtz problem shows that this novel approach results in robust and effective performance.

Structural topology optimization in the context of material degradation and fracture has been gaining considerable interest. In this work we explore the use of the phase field method for fracture in order to increase the fracture resistance, or strength, of a structure prior to failure by directly constraining the phase field approximation of the fracture surface energy. A density-based topology optimization framework is used and the total weight of the structure is minimized. The analytical sensitivities are derived and an efficiency gain based on the Schur-complement is presented for the computation of the sensitivities. The effectiveness of the proposed method is then demonstrated for two benchmark examples.

A computational model for simulations of fused filament fabrication (FFF) – often also called fused deposition modeling (FDM) – that includes viscoelastic stresses, is presented. The flow of a viscoelastic polymer filament from a nozzle, the deposition of the filament onto a substrata or previously deposited filaments, and the cooling and solidification of the filament, are captured using the one-fluid formulations, where a single set of equations is written for all the fluids and phases involved (polymer and air). The governing equations are solved on a regular stationary grid and the boundary between the polymer and the air is followed using connected marker points. The model is based on an earlier development where the polymer stresses were assumed to be Newtonian. The method is tested by simulating the extrusion of a polymer from a nozzle and the predicted swelling compared with results from the literature. The method is then used to simulate three filaments deposited on top of each other and the fabrication of an object consisting of a filament deposited perpendicularly on top of several filaments. The effect of including viscoelasticity on the extruded filament size and shape are discussed.

This paper presents a new formulation of the coupled reduced-order modeling technique for fluid–structure interaction problems. The problem addressed here is a classical vibro-acoustic issue, which is a coupled vibration of an acoustic fluid in an elastic structure . Discretization of the problem yields a model having many degrees of freedom, which may impede rapid simulation and analysis. Projection-based model reduction is thus the most popular way to handle this problem. Conventionally, structure and fluid modes are independently employed to reduce their own degrees of freedom, and the Schur complement is then used to make a weak coupling between the two domains. In this work, we suggest a new coupled formulation to build a strong connection between the fluid and structure, which is mathematically a sequential projection from structure to fluid. The proposed strongly coupled formulation provides insight into the way that the structural vibration energy is transmitted to the fluid domain. Consequently, it can offer more precise reduced-order modeling of the fluid–structure interaction problems than conventional approaches. Numerical results herein demonstrate improved accuracy of the proposed method.

We extend a recently-developed framework for isogeometric analysis of composite laminates to drive material damage evolution with a smoothed strain field. This builds on ideas from gradient-enhanced continuum damage modeling, and is intended to limit the dependence of damage predictions on the choice of discrete mesh. The resulting enhanced framework models each lamina of a composite shell structure as a Kirchhoff–Love thin shell. To account for the anisotropic damage modes of laminae, we smooth a tensor-valued strain by solving an elliptic partial differential equation (PDE) system on each lamina. This strain-smoothing PDE system is formulated to be independent of the choice of coordinates and is applicable to general manifold shell geometries. Numerical examples illustrate the enhanced damage model’s validity, mesh-independence, and applicability to complex industrial geometries.

This work makes a first attempt to conduct a geometrically non-linear analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) structures, with surface-bonded active layers, based on Kirchhoff shell theory. Uniform and three different distribution types of functionally graded nanocomposites are presented. These distributions are assumed to be uniaxially aligned in the axial direction and functionally graded in the shell thickness direction, while the electric potential through the thickness of the active layers is assumed to be linear. A comparison study is carried out in order to show the applicability of the current formulation applied to various shapes of shell structures . The effects of various parameters as volume fraction, distribution of nanotubes , geometrical characteristics as well as load boundary on non-linear behavior of the smart FG-CNTRC structures are investigated.

In the present study, natural convection heat transfer of Al 2 O 3 -water nanofluid inside a square cavity with a solid circular cylinder is investigated numerically. For numerical computations, the finite element method is used by taking into consideration Buongiorno’s two-phase model. Parts of the vertical surfaces of cavity are kept at constant temperature (left wall T h and right wall T c ) while the other walls (horizontal walls and the remaining of the vertical walls) are taken as adiabatic. The effects of some pertinent parameters such as the Rayleigh number ( 1 0 3 ≤ R a ≤ 1 0 6 ) , nanoparticle volume fraction ( 0 ≤ ϕ ≤ 0 . 04 ), thermal conductivity of the solid cylinder ( k w = 0 . 28 , 0.76, 1.95, 7 and 16), radius of solid cylinder ( 0 . 1 ≤ R ≤ 0 . 4 ), heat source/sink length ( 0 . 2 ≤ D ≤ 0 . 8 ), and the heat source/sink position ( 0 . 2 ≤ B ≤ 0 . 8 ) on the fluid flow and heat transfer characteristics are investigated. The obtained numerical results are depicted graphically and discussed in detail from the point of view of the streamlines, isotherms, nanoparticle volume fractions and the local and average Nusselt number N u . It is indicated that the heat transfer is enhanced with an increase in the nanoparticle volume fraction for all studied Rayleigh numbers. Furthermore, the thermal conductivity, solid circular cylinder size, D and B parameters are the key factors to control and optimize the heat transfer inside the cavity that is partially heated and cooled. The proposed method is found to be in good agreement between previously published experimental and numerical results.

Partially supported by the National Science Foundation (NSF), United States , Grants CMMI-1235238 and CMMI-1351742. The first and second authors would also like to acknowledge UP postdoctoral fellowship program and the Eskom Power Plant Engineering Institute (EPPEI).

In this work, we present a constrained batch-parallel Bayesian optimization (BO) framework, termed pBO-2GP-3B, to accelerate the optimization process for high-dimensional and computationally expensive problems, with known and unknown constraints. Two Gaussian processes (GPs) are simultaneously constructed: one models the objective function, whereas the other models the unknown constraints. The known constraint is penalized directly into the acquisition function. For every iteration, three batches are built in sequential order: the first two are the acquisition hallucination and the exploration batches for the objective GP, respectively, and the third one is the exploration batch for the classification GP. The pBO-2GP-3B optimization framework is demonstrated with three synthetic examples (2D and 6D), as well as a 33D multi-phase solid–liquid computational fluid dynamics (CFD) model for the design optimization of a centrifugal slurry pump impeller.

We present and analyze a semi-implicit finite element method (FEM) for the dynamical incompressible magnetohydrodynamics (MHD) equations. The finite element approximation is based on mixed conforming elements, where Taylor–Hood type elements are used for the Navier–Stokes equations and Nedelec edge elements are used for the magnetic equation. The divergence free conditions are weakly satisfied at the discrete level. Due to the use of Nedelec edge element, the proposed method is particularly suitable for problems defined on non-smooth and multi-connected domains. For the temporal discretization, we use a semi-implicit scheme which only needs to solve a linear system at each time step. Moreover, the linearized mixed FEM is energy conserving. We establish an optimal error estimate under a very low assumption on the exact solutions and domain geometries. Numerical results are provided to show its effectiveness and verify the theoretical analysis. In particular, a benchmark lid-driven cavity problem is provided to show that the proposed numerical method produces results as good as that of the ones which are divergence free.

Landslide dynamics are characterized by a phase transition from a solid like behavior to a fluid one. Since the material is moving through space special discretization techniques are required. Four-dimensional space–time finite elements for fluid–structure interactions are applied together with the single-phase level-set method to investigate the dynamics of 3D landslides interacting with flexible walls. The wall is modeled as a geometric nonlinear solid with elastic–viscoplastic material behavior , whereas the liquefied soil is described with the Navier–Stokes equations for incompressible Bingham fluids. Between fluid and solid advanced coupling conditions are taken into account incorporating friction. In order to solve the governing equations of the multi-field problem, the weighted-residuals method is applied, which is discretized by time-discontinuous space–time elements. Within the monolithic solution procedure for the coupled problem, the kinematics of both solid and fluid are described using velocities as primary variables. A pseudo-structure adapts the coordinates of the fluid mesh to the deformations of adjacent structures. The single-phase level-set method describes the motion of the free surface of the sliding material. To accurately transport the free surface a pde-based extrapolation is used for the velocities. Various effects in 3D landslide dynamics are investigated.

We are concerned with optimization of macroscale elastic structures that are designed utilizing spatially varying microscale metamaterials. The macroscale optimization is accomplished using gradient-based nonlinear topological optimization. But instead of using density as the optimization decision variable, the decision variables are the multiple parameters that define the local microscale metamaterial. This is accomplished using single layer feedforward Gaussian basis function networks as a surrogate models of the elastic response of the microscale metamaterial. The surrogate models are trained using highly resolved continuum finite element simulations of the microscale metamaterials and hence are significantly more accurate than analytical models e.g. classical beam theory . Because the derivative of the surrogate model is important for sensitivity analysis of the macroscale topology optimization , a neural network training procedure based on the Sobolev norm is described. Since the SIMP method is not appropriate for spatially varying lattices , an alternative method is developed to enable creation of void regions. The efficacy of this approach is demonstrated via several examples in which the optimal graded metamaterial outperforms a traditional solid structure.

The aim of the paper is to show that the isogeometric Boundary Element Method (isoBEM) has advantages over other numerical methods when applied to problems in potential flow. The problems presented here range from the flow past an obstacle to confined and unconfined seepage problems in isotropic and anisotropic media. It is shown how accurate results can be obtained with very few unknowns. The superior capability of NURBS to describe geometry and the variation of the unknowns is exploited.

Phase-field models for fracture problems have attracted considerable attention in recent years, which are capable of tracking the discontinuities numerically, and also produce complex crack patterns in many applications. In this paper, a phase-field model for a general nonlinearly elastic material is proposed using a novel additive decomposition of strain energy. This decomposition has two parts: one is principal stretch related and the other solely composed of volumetric deformation, which accounts for different behaviors of fracture in tension and compression. We construct the Lagrangian by integrating the split energies and the separation energy from phase-field approximation for discrete cracks. A coupled system of equations is also derived that governs the deformation of the body and the evolution of phase field. The capability and performance of the proposed model are demonstrated in several representative examples. Our results show that the predicted fracture surfaces are in good agreement with experimental observations. Compared with the previous models in which the energy is simply split into the isochoric and volumetric parts, the present model is numerically more robust and effective in simulating sharp cracks. The present model can also aid researchers to control the degree of tension–compression asymmetry in the nonlinear regime of deformation, which can be naturally extended to simulate the fracture of the rubber-like materials with tension–compression asymmetry.

2018 marked the 80th anniversary of Lanczos (1938) and 2017 was the 50th anniversary of Villadsen and Stewart (1967), both seminal works in development of the Orthogonal Collocation Method, also known as the Pseudospectral Method and Differential Quadrature Method . Due to these milestones, a retrospective on development of the method seems appropriate. The method’s development as a Method of Weighted Residuals (MWR), with numerical integration, is reviewed. It is placed into historical perspective and developments in disparate disciplines are reconciled. We show the equivalence of methods that were thought by many to be different. Guidance in navigating the minefield of terminology is also provided. In addition to reviewing the development of the method, we show some refinements which are not known to most practitioners. Chief amongst these are the correct treatment of boundary conditions involving fluxes. We demonstrate that a proper treatment of flux boundary conditions can sometimes reduce errors by several orders of magnitude.

This paper presents numerical studies with three classes of quadratic Generalized FEM (GFEM) approximations and shows that all of them lead to errors that are orders of magnitude smaller than the FEM with quarter-point elements, provided that appropriate enrichments are selected. However, all of them lead to severely ill-conditioned systems of equations, with a condition number up to O ( h − 10 ) in the case of the GFEM based on a quadratic partition of unity. Enrichment modifications able to address the ill-conditioning of quadratic GFEM approximations while preserving their optimal convergence are proposed. A robust enrichment modification strategy based on a discontinuous FE interpolant is proposed to control the conditioning of branch function enrichment. The discontinuous FE interpolant is a generalization of the continuous one used with the Stable GFEM (SGFEM). We show that SGFEM spaces based on p-hierarchical FEM enrichments are the same as their GFEM counterparts. This guarantees that both GFEM and SGFEM spaces will lead to the same solution. This is not the case for the other classes of second-order spaces. The robustness of the proposed approximation spaces with respect to the crack location in the mesh is also demonstrated.

Mechanical Shape Correlation (MSC) is a novel Integrated Digital Image Correlation (IDIC) based technique used for parameter identification. Digital images taken during an experiment are correlated and coupled to a Finite Element model of the specimen, in order to find the correct parameters in this numerical model. In contrast to regular IDIC techniques , where the images consist of a grayscale speckle pattern applied to the sample, in MSC the images are projections based on the contour lines of the test specimen only. This makes the technique suitable in cases where IDIC cannot be used, e.g., when large deformations and rotations cause parts of the sample to rotate in or out-of-view, or when the speckle pattern degrades due to large or complex deformations, or when application of the pattern is difficult because of small or large specimen dimensions . The method targets problems for which the outline of the specimen is large with respect to the volume of the structure and changes significantly upon deformation. The technique is here applied to virtual experiments with stretchable electronic interconnects, for identification of both elastic and plastic properties. Furthermore, attention is paid to the influence of algorithmic choices and experimental issues. The method reveals good convergence and adequate initial guess robustness. The results are promising and indicate that the method can be used in cases of either large, complex or three-dimensional displacements and rotations on any scale.

This work presents a generalized topology optimization approach for the design of hierarchical lattice structures with the development of an Approximation of Reduced Substructure with Penalization (ARSP) model. The structure is assumed to be composed of substructures with a common lattice geometry pattern. Unlike conventional homogenization-based designs assuming the separation of scales, this work considers two different yet connected scales. Each substructure is condensed first into a super-element with a reduced degrees of freedom and is associated with a density design variable indicating the material volume fraction. The density design variable is linked to a lattice geometry feature parameter. A surrogate model is particularly built with the aid of proper orthogonal decomposition and diffuse approximation, mapping the density to super-element stiffness matrix. The derivative of super-element matrix with respect to the associated density can therefore be evaluated efficiently and explicitly. The super-element matrix is further augmented with a penalized density to control the structural complexity. The optimality criteria method is used for the update of design variables. Numerical examples show that both the size and the lattice pattern of substructure have essential influences on the design, indicating the necessity of performing connected hierarchical modeling and design.

We study in this paper the accuracy and stability of partially and fully implicit schemes for phase field modeling. Through theoretical and numerical analysis of Allen–Cahn and Cahn–Hilliard models, we investigate the potential problems of using partially implicit schemes, demonstrate the importance of using fully implicit schemes and discuss the limitation of energy stability that are often used to evaluate the quality of a numerical scheme for phase-field modeling. In particular, we make the following observations: 1. a convex splitting scheme (CSS in short) can be equivalent to some fully implicit scheme (FIS in short) with a much different time scaling and thus it may lack numerical accuracy; 2. most implicit schemes are energy-stable if the time-step size is sufficiently small; 3. a traditionally known conditionally energy-stable scheme still possesses an unconditionally energy-stable physical solution; 4. an unconditionally energy-stable scheme is not necessarily better than a conditionally energy-stable scheme when the time step size is not small enough; 5. a first-order FIS for the Allen–Cahn model can be devised so that the maximum principle will be valid on the discrete level and hence the discrete phase variable satisfies | u h ( x ) | ≤ 1 for all x and, furthermore, the linearized discretized system can be effectively preconditioned by discrete Poisson operators.

Straight beams, rods and trusses are common elements in structural and mechanical engineering, but recent advances in additive manufacturing now also enable efficient freeform fabrication of curved, deformable beams and beam structures, such as microstructures, metamaterials and conformal lattices. To exploit this new design freedom for applications with nonlinear mechanical behavior, we introduce an isogeometric method for shape optimization of curved 3D beams and beam structures. The geometrically exact Cosserat rod theory is used to model nonlinear 3D beams subject to large deformations and rotations. The initial and current geometry are parameterized in terms of NURBS curves describing the beam centerline and an isogeometric collocation approach is used to discretize the strong form of the balance equations. Then, a nonlinear optimization problem is formulated in order to optimize the positions of the control points of the NURBS curve that describes the beam centerline, i.e., the geometry or shape of the beam. To solve the design problem using gradient-based algorithms, we introduce semi-analytical, inconsistent analytical and fully analytical approaches for calculation of design sensitivities. The methods are numerically validated and their performance is investigated, before the applicability and versatility of our 3D beam shape optimization method is illustrated in various numerical applications, including optimization of an auxetic 3D metamaterial.

A non-intrusive B-Splines Bezier Elements based Method (BSBEM) is proposed as an efficient tool for uncertainty propagation analysis in physical hyperbolic problems. The model’s output response is approximated using a surrogate model whose coefficients are obtained from a set of deterministic calls by means of a regression technique . The accuracy, efficiency and the generality of the proposed approach are assessed using benchmark numerical examples by comparing the convergence of the statistics moments with those of the Polynomial chaos-based point collocation method (Pcol) and the Monte Carlo (MC) method. The generic character of the proposed approach allows it to be implemented for several engineering fields. BSBEM is applied to quantify uncertainty propagation through dam break flows modelled by shallow water equations. The obtained results, which are depicted in terms of water depth and inundation line confidence intervals, show that with a meticulous exploitation of the multi-element aspect and the smoothness feature of the basis functions, the proposed method provides an accurate and smooth approximation of the stochastic output response.

A mixed finite element formulation for compressible finite hyperelasticity, possibly with strongly deformation dependent stiffening fibre reinforcement, is developed and verified. Reinforcement by two, mechanically equivalent, in general oblique, fibre families is considered. In the provided numerical experiments, residual based error estimation is essential for a simple h-adaptive mesh refinement strategy. It is shown to be effective to resolve stress concentrations, also in combination with stress oscillations (checker boarding) in the presence of strong anisotropy especially when mixed element constructs with enriched polynomial order are used. The novel mixed generalised (Lagrangian) metric based formulation is illustrated using a simple but fully coupled material model. The ground substance is the common so-called compressible neo-Hooke model and the standard reinforcement part is augmented by second order anisotropic invariants. The theoretical formulation admits approaching the inextensible limit seamlessly, by-passing the use of a separate decoupled material model. The Hu–Washizu implementation used here relies on a certain extensibility for elimination of the auxiliary variables, approximated in L 2 at the element level. The condensation yields a purely H 1-based displacement formulation at the global level.

The main aim of this research work is to find the numerical solution based on a meshless technique for both the time-dependent Cahn–Hilliard and tumor growth partial differential equations. The temporal variable is discretized using a second-order method based on semi-implicit backward differential formula, and the stabilized term is added to the considered time discretization. Also, an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization. Besides, to approximate the spatial variables, the element-free Galerkin method is considered for both mathematical models. The result of full discrete schemes in studied models is solved via Biconjugate gradient stabilized algorithm. Some numerical simulations are reported to show the capability of the numerical scheme presented here.

This paper presents a new approach for level set based shape optimization using trimmed hexahedral meshes that geometrically conform to shape and topology changes during the optimization process. The marching cubes algorithm is employed to generate trimmed hexahedral meshes at each optimization iteration by splitting a regular background hexahedral mesh with the zero-isosurface of a level set function . A simple method is proposed to handle nonconforming polyhedral elements at the interface due to the face ambiguity arising in the marching cubes algorithm. The motion of the zero-isosurface of a level set function is obtained from the solution of the Hamilton–Jacobi transport equation using a normal velocity field defined on the regular background hexahedral mesh. Numerical results show that the present approach provides a natural and effective tool to solve shape optimization problems relying on explicitly discretized meshes of the shapes throughout the optimization process.

Coronary artery bypass graft surgery (CABG) is performed on more than 400,000 patients annually in the U.S. However, saphenous vein grafts (SVGs) implanted during CABG exhibit poor patency compared to arterial grafts, with failure rates up to 40% within 10 years after surgery. Differences in mechanical stimuli are known to play a role in driving maladaptation and have been correlated with endothelial damage and thrombus formation. As these quantities are difficult to measure in vivo, multi-scale coronary models offer a way to quantify them, while accounting for complex coronary physiology. However, prior studies have primarily focused on deterministic evaluations, without reporting variability in the model parameters due to uncertainty. This study aims to assess confidence in multi-scale predictions of wall shear stress and wall strain while accounting for uncertainty in peripheral hemodynamics and material properties. Boundary condition distributions are computed by assimilating uncertain clinical data, while spatial variations of vessel wall stiffness are obtained through approximation by a random field. We developed a stochastic submodeling approach to mitigate the computational burden of repeated multi-scale model evaluations to focus exclusively on the bypass grafts. This produces a two-level decomposition of quantities of interest into submodel contributions and full model/submodel discrepancies. We leverage these two levels in the context of forward uncertainty propagation using a previously proposed multi-resolution approach. The time- and space-averaged wall shear stress is well estimated with a coefficient of variation of

Stress integration algorithm based on finite difference method (FDM) was proposed to effectively deal with both first and second derivatives of yield and potential functions which are the lengthiest component in stress integration procedure. With the proposed numerical algorithm, both first and second derivatives of yield function are approximated by central difference method, so that finite element modeling using advanced constitutive model could be easily performed no matter how complicated its derivatives are. For the verification purpose, the algorithm was applied for advanced constitutive models: Plastic anisotropy model under associated (AFR) and non-associated flow rule (non-AFR), the homogeneous anisotropic hardening (HAH) model under associated (AFR) and non-associated flow rule (non-AFR). The proposed algorithm was verified with single element loading–unloading and cup-drawing simulations. The Euler backward method based on both the proposed numerical algorithm and analytical derivatives were employed for verification purpose. The accuracy and time efficiency of the proposed numerical algorithm were evaluated by comparing the simulation results from analytical derivatives. In addition, the applicability of the proposed numerical algorithm for the HAH models was estimated with single element loading–unloading, loading–reloading, and deep-drawing/springback simulations. Non-associated flow plasticity for the HAH model is newly proposed to improve numerical efficiency with finite difference method by keeping the same level of accuracy as associated flow rule plasticity. All the simulation results proved that the proposed numerical algorithm can be widely used for the implementation of advanced constitutive models.

As a soft material with an almost negligible bending stiffness , a membrane may easily lose its mechanical stability. To capture its entire instability process, intensive computation is required, especially in the case of short wave length . The objective of this paper is to construct an efficient model to simulate and study the instability phenomena of circular membranes . By using the method of Fourier series with slowly variable coefficients in the circumferential direction , a new family of one-dimensional reduced finite elements are developed to study the three-dimensional problems. The nonlinear system is solved by the Asymptotic Numerical Method(ANM), which is reliable and efficient for tracing the bifurcation points and post-buckling equilibrium path compared with other classical non-linear solution algorithms . The accuracy and efficiency of the reduced model is verified by simulating the instability phenomena in stretched annular membranes and a compressed circular plate. The relation between critical loads and bifurcation patterns and the evolution of stress components in the entire wrinkling process are discussed. This study provides new simulation schemes to explore the instability in circular membranes under complex loadings and boundary conditions.